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Disturbance suppression beyond Nyquist frequency in hard disk drives
Mechatronics 20 (2010) 67–73
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Disturbance suppression beyond Nyquist frequency in hard disk drives Takenori Atsumi * Central Research Laboratory, Research and Development Group, Hitachi Ltd., 1 Kirihara, Fujisawa, Kanagawa 252-8588, Japan
a r t i c l e
i n f o
Keywords: Mechatronic systems Tracking Disturbance rejection Vibration and modal analysis Design methodologies
a b s t r a c t In conventional hard disk drives, a control system compensates for vibration in which the frequency is higher than the Nyquist frequency by using a multi-rate filter that decreases the gain above the Nyquist frequency. However, such a control system can only avoid instability and cannot suppress disturbances above the Nyquist frequency. To overcome this problem, a control system design method that suppresses disturbances beyond the Nyquist frequency is proposed. This method uses frequency responses of a controlled object and a digital controller to calculate the gain of the sensitivity function in a sampled-data system without requiring complex calculations involving matrices, and realizes a stable resonant filter that decreases the gain of the sensitivity function above the Nyquist frequency. Significant suppression of the vibrations caused by the disturbances beyond the Nyquist frequency is demonstrated by implementing this method in the head-positioning system of a hard disk drive. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The head-positioning accuracy of hard disk drives must be improved to meet the increasing demand for larger storage capacity [1,2]. In the head-positioning system of a hard disk drive, vibrations of the mechanical system impair positioning accuracy. Therefore, it is important to compensate for the vibrations in the headpositioning system to improve positioning accuracy [3–6]. The head-positioning system of a hard disk drive is implemented in a sampled-data control system that has a sampler and a hold. The head-positioning system also has mechanical vibrations whose frequencies exceed the Nyquist frequency of the control system. As such, the head-positioning control system needs to compensate for these mechanical vibrations beyond the Nyquist frequency. The control system in conventional hard disk drives compensates for these vibrations by using a multi-rate filter that decreases the gain above the Nyquist frequency [7]. However, such control systems can only avoid the instability caused by mechanical resonances; they cannot actually suppress the vibrations. Many studies have been done on control systems that improve the control performance on the basis of sampled-data control theories and multi-rate control techniques [8–18]. However, these studies do not focus on how to suppress the vibrations with frequencies higher than the Nyquist frequency. To overcome the above-mentioned problems, we have studied design methods of a sampled-data control system based on frequency responses [19]. In this paper, a control system that sup* Tel.: +81 466 98 4166. E-mail addresses: [email protected], [email protected]. 0957-4158/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2009.06.002
presses disturbances beyond the Nyquist frequency is proposed. The design method for the multi-rate control system uses frequency responses of a controlled object and a digital controller for calculations of the gain of the sensitivity function in a sampled-data system without requiring complex calculations involving matrices. By implementing the method in the headpositioning system of a hard disk drive, it is confirmed that the control system can suppress the vibrations caused by the disturbances beyond the Nyquist frequency.
2. Head-positioning system of hard disk drives A hard disk drive consists of a voice coil motor (VCM), several magnetic heads, several disks, and a spindle motor, as shown in Fig. 1. The head-positioning control system in hard disk drives is illustrated in Fig. 2. In the head-positioning system, the control variable is the head-position signal, which is embedded in disks and is read by magnetic heads. This means the head-position signal is only available as a discrete-time signal. The control input is the voltage supplied to a power amplifier that drives the VCM and magnetic heads, which is calculated by a digital signal processor at certain intervals. Therefore, a head-positioning control system can be thought of as a sampled-data control system that has a sampler and a hold. A block diagram of the sampled-data control system with a multi-rate digital filter is shown in Fig. 3. Here, S is the sampler, C½z is the single-rate digital controller, Ip ½z is the interpolator, F m ½z is the multi-rate digital filter, Hm ðsÞ is the multi-rate hold, Pc ðsÞ is the continuous-time plant, r is the reference signal, ud is the output signal from C½z; uc is the output signal from Hm ðsÞ; dc
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T. Atsumi / Mechatronics 20 (2010) 67–73
tions and the characteristics of disturbance rejections in a sampled-data system by using frequency responses. The relationship between a continuous-time signal and a discrete-time signal is defined in this paper. For illustration purposes, the sinusoidal wave x0 ðtÞ is defined as
x0 ðtÞ ¼ ejx0 t :
ð1Þ
The discrete-time signal xd ½n, which corresponds to the signal after sampling of x0 ðtÞ with the sampling time T s , can be given as
xd ½n ¼ x0 ðnT s Þ ¼ ejx0 T s n : Fig. 1. Primary components of HDD.
ð2Þ
The continuous-time signal xc ðtÞ, which is created from xd ½n, can be given as
xc ðtÞ ¼
1 X
dðt lT s Þejx0 t ¼
l¼1
1 jðx0 þxs lÞt um1 ; l¼1 e Ts
ð3Þ
where d is a Dirac delta function. Eqs. (2) and (3) indicate that the discrete-time signal caused by a single complex sinusoid can be given by a single frequency, and the continuous-time signal caused by the discrete-time signal consists of multiple frequencies.
3.1. Analysis of sensitivity function
Fig. 2. Head-positioning system.
is the disturbance signal in continuous time, yc0 is the output signal from P c ðsÞ; yc is the head-position signal in continuous time, and yd is the head-position signal in discrete time. In this study, the multirate number is m, the sampling time is T s [s], and the sampling frequency is xs ¼ 2p=T s ½rad=s. The timing used in reading or writing user data in hard disk drives corresponds to intersampling in a sampled-data control system. In other words, the essential purpose of a head-positioning system is to control the maximum displacement of the control variable including intersampling behaviors. Therefore, the control system should be designed as a sampled-data system so that the controller compensates for the intersampling vibrations. The head-positioning system has two control modes; one is ‘‘seek control,” which moves the head onto the target track, and the other is ‘‘following control,” which maintains the head position on the target track. In the seek control, the main target of the control system is the transient characteristics of the head position, but in the following control, the main target is the steady-state characteristics. The purpose of this study is to develop a following control system, and so from here on, the focus of this paper is the steadystate characteristics of the head-position control. 3. Analysis of sampled-data control system using frequency response
In some papers, the gain-frequency response of the sensitivity function in the sampled-data system is defined by using the worst-case H1 norm of systems or the upper/lower bound [24,25]. However, obviously, a head-positioning system has to control the maximum displacement of the control variable, and the maximum displacement has no bound in the steady-state characteristics. Therefore, the infinity norms of signals are used for defining the gain of the sensitivity function for head-positioning control in hard disk drives. In this paper, a gain-frequency response of the sensitivity function in a sampled-data control system jSsd j at x0 is given by the following equation using dc and yc in Fig. 3:
jSsd ðjx0 Þj ¼
Fig. 3. Block diagram of sampled-data control system.
ð4Þ
where dc ðt; x0 Þ is a single complex sinusoid,
dc ðt; x0 Þ ¼ ejx0 t :
ð5Þ
The transfer characteristics from ud to yd at x0 , which is a controlled object in a discrete-time system, is given as Pd ðjx0 Þ,
Pd ðjx0 Þ ¼
1 1 X Wðjx0 þ jxs kÞPc ðjx0 þ jxs kÞ; T s k¼1
ð6Þ
where
WðjxÞ ¼ Ip ½ejxT s =m F m ½ejxT s =m Hm ðjxÞ:
ð7Þ
In a steady-state with dc ðt; x0 Þ ¼ ejx0 t ; ud can be given by the following equation:
ud ½n ¼ To avoid complex calculations involving matrices, the proposed method uses frequency responses to analyze the sampled-data control system [20–23]. The method handles intersampling vibra-
kyc ðtÞk1 ¼ sup jyc ðtÞj; kdc ðt; x0 Þk1 t
C½ejx0 T s ejx0 T s n : 1 þ Pd ðjx0 ÞC½ejx0 T s
ð8Þ
Here, uc ðtÞ and yc0 ðtÞ consist of multiple frequencies and are given by
uc ðtÞ ¼
1 X C½ejx0 T s 1 Wðjx0 þ jxs lÞejðx0 þxs lÞt ; j x T 1 þ Pd ðjx0 ÞC½e 0 s T s l¼1
yc0 ðtÞ ¼
1 X C½ejx0 T s 1 Wðjx0 þ jxs lÞP c ðjx0 þ jxs lÞejðx0 þxs lÞt : 1 þ P d ðjx0 ÞC½ejx0 T s T s l¼1
ð9Þ
ð10Þ
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T. Atsumi / Mechatronics 20 (2010) 67–73
Therefore, yc ðtÞ also consists of multiple frequencies and is given by
yc ðtÞ ¼ yc0 ðtÞ þ dc ðt; x0 Þ 1 X Kðx0 ; lÞejðx0 þxs lÞt ; ¼ Cðx0 Þejx0 t þ
F m ½z ¼ 1 þ F r ½z; ð11Þ
l¼1;–0
CðxÞ ¼ 1
1 C½ejxT s WðjxÞP c ðjxÞ ; T s 1 þ P d ðjxÞC½ejxT s
ð12Þ
1 C½ejxT s Wðjx þ jxs lÞPc ðjx þ jxs lÞ Kðx; lÞ ¼ : Ts 1 þ Pd ðjxÞC½ejxT s
ð13Þ
Finally, jSsd ðjx0 Þj can be given by
t
1 X
jKðx0 ; lÞj:
l¼1;–0
¼
1 X
1þ
!
Kðx0 ; lÞ ejx0 T s n !
Kðx0 ; lÞ ejx0 T s n
1 ejx0 T s n ¼ Sd ðjx0 Þejx0 T s n ; 1 þ P d ðjx0 ÞC½ejx0 T s
5.1. Controlled object
ð15Þ
Pc0 ðjxÞ : jxj 2xs ;
ð23Þ
0 : jxj > 2xs :
Pc0 ðsÞ ¼ Ps ðsÞ Pdl ðsÞ:
ð24Þ
The mechanical model Ps is given by follows [3]:
Ps ðsÞ ¼ K p
w X
ak ðlÞ
l¼1
s2 þ 2fk ðlÞxnk ðlÞs þ xnk ðlÞ2
;
ð25Þ
ð16Þ
Therefore, the control system can suppress the vibration with frequency x0 when
ð17Þ
and 1 X
jWðjx0 þ jxs lÞP c ðjx0 þ jxs lÞj:
20
Gain [dB]
jSd ðjx0 Þj < 1;
jWðjx0 ÞPc ðjx0 Þj
Pc0 is assumed to be given by the product of the ‘‘mechanical model Ps ” and the ‘‘equivalent delay-time model P dl .”
where Sd ðjx0 Þ is the sensitivity function at x0 in a discrete-time system. P1 Eq. (15) indicates that when l¼1;–0 jKðx0 ; lÞj is sufficiently small, Ssd can be given by
Ssd ðjx0 Þ ’ Sd ðjx0 Þ:
The sampling time T s of the hard disk drive used in this study was 76:7 ls (xs ¼ 2p 13000 ½rad=s and the Nyquist frequency was 6500 Hz). The frequency of disturbance xr is set to 7000 Hz. The frequency response of the mechanical system is plotted by the dashed line in Fig. 4.
Pc ðjxÞ ¼
l¼1
¼
ð22Þ
It is assumed that jPc ðjxÞj is vanishingly small when x > 2xs . Thus, to reduce the calculation load, the frequency response of Pc ðjxÞ is defined as
Kðx0 ; lÞejðx0 T s þ2plÞn
l¼1;–0 1 X
Here, the disturbance frequency is defined as xr . To suppress the disturbance, F r should have stable resonant characteristics at xr , and the control system, which includes F r , should satisfy the following equation [26,27]:
5. Control system design
To suppress vibration with frequency x0 , the control system should have a certain level of jWP c j at x0 so that jCðx0 Þj becomes small and a small amount of jWP c j at the aliasing frequencies so P1 that l¼1;–0 jKðx0 ; lÞj becomes small. On the other hand, yd ½n can be given by the following equation:
Cðx0 Þ þ
ð21Þ
ð14Þ
3.2. Disturbance suppression in sampled-data system
1 X
ð20Þ
where
\½Pd ðjxr ÞC½ejxr T s ¼ \½1 þ P d0 ðjxr ÞC½ejxr T s :
l¼1;–0
yd ½n ¼ yc ðnT s Þ ¼ Cðx0 Þejx0 T s n þ
1 1 X W 0 ðjx0 þ jxs kÞP c ðjx0 þ jxs kÞ; T s k¼1
W 0 ðjxÞ ¼ Ip ½ejxT s =m Hm ðjxÞ:
and
jSsd ðjx0 Þj ¼ sup jyc ðtÞj ¼ jCðx0 Þj þ
ð19Þ
where F r is the multi-rate resonant filter. When F r ½z ¼ 0, the transfer characteristics from ud to yd at x0 are given by follows:
Pd0 ðjx0 Þ ¼
where
¼
The multi-rate filter F m is given by the following equation:
0 −20 Measurement data Mathematical model
ð18Þ −40
l¼1;–0
10
3
4
10
Frequency [Hz]
4. Disturbance suppression using multi-rate resonant filter Phase [deg.]
To suppress the disturbance beyond the servo bandwidth of the control system, the control system should have stable resonant characteristics at the disturbance frequency [4,26,27]. However, (18) indicates that the control system should have small gains at aliasing frequencies of the disturbance frequency. Therefore, when the disturbance frequency is higher than the Nyquist frequency, the control system should realize the resonant characteristics by using continuous-time filters or multi-rate filters. In this paper, a design method that uses the multi-rate filter is used.
0 −90 −180 −270
Measurement data Mathematical model
−360 10
3
4
10
Frequency [Hz] Fig. 4. Frequency response of mechanical system.
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T. Atsumi / Mechatronics 20 (2010) 67–73
where w is the number of modes, ak is the residue of each mode, xnk and fk are the natural frequency [rad/s] and the damping ratio of the resonance, respectively, and K p is the plant gain. These parameters were fixed so that the model’s frequency response coincided with the measured result indicated by the dashed line in Fig. 4. To be more precise, w was set to seven (the model order is fourteen), K p to 4:3 107 , and the values of the other parameters are listed in Table 1. The solid line in Fig. 4 represents the frequency response of this mathematical model. The equivalent delay-time model P dl , which includes the characteristics of the delay-time element, is given by follows [20]:
Pdl ðsÞ ¼ ess ;
ð26Þ
where s is the equivalent dead time. Here, used in this study is 10 ls.
s of the hard disk drive
trolled object Pd0 in which F m ½z ¼ 1. The frequency response of Pd0 is indicated by the dashed line shown in Fig. 5. To make the phase margin more than 30°, the gain margin more than 4.5 dB, and the open-loop gain 0-dB crossover frequency 1100 Hz, C is set according to the following equation:
C½z ¼
11:8ðz 0:908Þðz 0:953Þ : ðz 0:161Þðz 1Þ
The frequency response of Pd0 C (the open-loop characteristics without the resonant filter) is indicated by the dashed line in Fig. 6. To suppress the disturbance, the multi-rate filter F m should have stable resonant characteristics at xr . From (19) and (22), the phase of the multi-rate resonant filter F r at the disturbance frequency xr should be
\F r ½ejxr T s =2 ¼ \Pc ðjxr Þ \Ip ½ejxr T s =2 \Hm ðjxr Þ \C½ejxr T s \½1 þ Pd0 ðjxr ÞC½ejxr T s :
5.2. Multi-rate hold and interpolator In this paper, the multi-rate number m is set to two. The multirate hold Hm is a zero-order hold (ZOH) and the transfer characteristic is given by the following equation:
1 eðT s =2Þs : s
ð30Þ
As a result, F m ½z was set as follows:
F m ½z ¼ 1 þ
0:0479ðz 4:37Þðz 1Þ : ðz2 þ 0:233z þ 0:998Þ
ð31Þ
ð27Þ
The interpolator consists of an up-sampler and an interpolation filter. The transfer function of the interpolation filter can be given by
Ip ½z ¼ 1 þ z1 :
ð28Þ
20
Gain [dB]
Hm ðsÞ ¼
ð29Þ
0 −20
Without resonant filter With resonant filter
5.3. Controller design −40
0
1000
2000
First, the single-rate controller C½z, which stabilizes the control system, is given by the ‘‘PI-lead filter” and designed for the con-
3000
4000
5000
6000
5000
6000
Frequency [Hz]
l
xnk ðlÞ
ak ðlÞ
fk ðlÞ
1 2 3 4 5 6 7
0 25,800 35,800 39,000 48,100 55,900 61,600
1.00 1.30 0.03 0.08 0.12 0.13 0.35
0 0.01 0.01 0.01 0.02 0.02 0.03
Phase [deg.]
0 Table 1 Parameters of Ps ðsÞ.
−90 −180 −270 −360
Without resonant filter With resonant filter 0
1000
2000
3000
4000
Frequency [Hz] Fig. 6. Frequency response of open-loop characteristics.
40
Gain [dB]
Gain [dB]
20 0 −20 −40
Without resonant filter With resonant filter 0
1000
2000
20 0
3000
4000
5000
6000
0
Frequency [Hz]
6000
8000
10000
8000
10000
0
−90
Phase [deg.]
Phase [deg.]
4000
Frequency [Hz]
0
−180 −270 −360
2000
Without resonant filter With resonant filter 0
1000
2000
3000
4000
5000
Frequency [Hz] Fig. 5. Frequency response of controlled object.
6000
−90 −180 −270 −360
0
2000
4000
6000
Frequency [Hz] Fig. 7. Frequency response of F m .
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T. Atsumi / Mechatronics 20 (2010) 67–73
jWðjxr ÞPc ðjxr Þj ¼ 3:11;
ð32Þ
15 10 5 0
Gain [dB]
The frequency response of F m is shown in Fig. 7, and the frequency response of P d is indicated by the solid line in Fig. 5. The characteristics of the control system with the resonant filter are explained below. The frequency response of the open-loop characteristics is plotted by the solid line in Fig. 6. The Nyquist diagram and the gain of the sensitivity function in discrete-time system Sd are shown in Figs. 8 and 9, respectively. These figures indicate that (17) is satisfied by using the resonant filter, and (18) is also satisfied because
−5 −10 −15
and 1 X
jWðjxr þ jxs lÞP c ðjxr þ jxs lÞj ¼ 0:11:
−20
ð33Þ
l¼1;–0
−25
By using the design results of the control system and (14), the gainfrequency responses of the sensitivity functions in sampled-data control systems were calculated. The frequency responses of jSsd j
−30 0
2000
4000
6000
8000
10000
12000
Frequency [Hz] Fig. 10. Frequency response of jSsd j.
8
are shown in Fig. 10. As seen from this figure, the control system suppresses the disturbance at 7000 Hz.
6
6. Simulation and experiment
Imaginary axis
4
2
0
−2
−4 −4
−2
0
2
4
6
8
10
Real axis Fig. 8. Nyquist diagram.
The simulation was performed for the control system, in which the frequency of disturbance signal dc is 7000 Hz, and the vibrational amplitude is 0.1 track. In Fig. 11, the dashed line represents the signal of yc without the resonant filter, and the solid line represents the signal of yc with the resonant filter. In Fig. 12, the dashed line represents the signal of yd without the resonant filter, and the solid line represents the signal of yd with the resonant filter. These figures indicate that the control system decreases vibrations in which the frequency is higher than the Nyquist frequency. To validate the effect of the proposed method, we did experiments using the conditions similar to those of the simulation. In these experiments, the sampling time of yc is 38:4 ls (1/2 of T s ) because the continuous-time signal cannot be measured in actual head-positioning systems. The time response of yc is shown in Fig. 13, and the time response of yd is shown in Fig. 14. In both figures, the dashed line represents the results without the resonant filter, and the solid line represents the results with the resonant
15
0.2
10
Without resonant filter With resonant filter
0.15
5 0.1
Position [track]
Gain [dB]
0 −5 −10 −15
0 −0.05 −0.1
−20
−0.15
−25 −30
0.05
0
2000
4000
6000
8000
Frequency [Hz] Fig. 9. Frequency response of jSd j.
10000
12000
−0.2
0
0.5
1
Time [ms] Fig. 11. Simulation results of yc .
1.5
2
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T. Atsumi / Mechatronics 20 (2010) 67–73
filter. These experimental results also indicate that the control system can decrease vibrations whose frequency is higher than the Nyquist frequency. The differences between the simulation result and the experimental result are caused by disturbances other than the disturbance signal dc in the actual head-positioning system.
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05
7. Conclusion
0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
1.5
2
Time [ms]
A control system design method that suppresses disturbances beyond the Nyquist frequency was proposed. This method uses frequency responses of a controlled object and digital controller for calculations of the gain of the sensitivity function in a sampleddata system without complex calculations involving matrices, and realizes a stable resonant filter that decreases the gain of the sensitivity function above the Nyquist frequency. By implementing this method in the head-positioning system of a hard disk drive, it is shown that the control system can significantly suppress the disturbance whose frequency was higher than the Nyquist frequency.
Fig. 12. Simulation results of yd .
References
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
1.5
2
Time [ms] Fig. 13. Experimental results of yc .
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
Time [ms] Fig. 14. Experimental results of yd .
1.5
2
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Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Disturbance suppression beyond Nyquist frequency in hard disk drives Takenori Atsumi * Central Research Laboratory, Research and Development Group, Hitachi Ltd., 1 Kirihara, Fujisawa, Kanagawa 252-8588, Japan
a r t i c l e
i n f o
Keywords: Mechatronic systems Tracking Disturbance rejection Vibration and modal analysis Design methodologies
a b s t r a c t In conventional hard disk drives, a control system compensates for vibration in which the frequency is higher than the Nyquist frequency by using a multi-rate filter that decreases the gain above the Nyquist frequency. However, such a control system can only avoid instability and cannot suppress disturbances above the Nyquist frequency. To overcome this problem, a control system design method that suppresses disturbances beyond the Nyquist frequency is proposed. This method uses frequency responses of a controlled object and a digital controller to calculate the gain of the sensitivity function in a sampled-data system without requiring complex calculations involving matrices, and realizes a stable resonant filter that decreases the gain of the sensitivity function above the Nyquist frequency. Significant suppression of the vibrations caused by the disturbances beyond the Nyquist frequency is demonstrated by implementing this method in the head-positioning system of a hard disk drive. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The head-positioning accuracy of hard disk drives must be improved to meet the increasing demand for larger storage capacity [1,2]. In the head-positioning system of a hard disk drive, vibrations of the mechanical system impair positioning accuracy. Therefore, it is important to compensate for the vibrations in the headpositioning system to improve positioning accuracy [3–6]. The head-positioning system of a hard disk drive is implemented in a sampled-data control system that has a sampler and a hold. The head-positioning system also has mechanical vibrations whose frequencies exceed the Nyquist frequency of the control system. As such, the head-positioning control system needs to compensate for these mechanical vibrations beyond the Nyquist frequency. The control system in conventional hard disk drives compensates for these vibrations by using a multi-rate filter that decreases the gain above the Nyquist frequency [7]. However, such control systems can only avoid the instability caused by mechanical resonances; they cannot actually suppress the vibrations. Many studies have been done on control systems that improve the control performance on the basis of sampled-data control theories and multi-rate control techniques [8–18]. However, these studies do not focus on how to suppress the vibrations with frequencies higher than the Nyquist frequency. To overcome the above-mentioned problems, we have studied design methods of a sampled-data control system based on frequency responses [19]. In this paper, a control system that sup* Tel.: +81 466 98 4166. E-mail addresses: [email protected], [email protected]. 0957-4158/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2009.06.002
presses disturbances beyond the Nyquist frequency is proposed. The design method for the multi-rate control system uses frequency responses of a controlled object and a digital controller for calculations of the gain of the sensitivity function in a sampled-data system without requiring complex calculations involving matrices. By implementing the method in the headpositioning system of a hard disk drive, it is confirmed that the control system can suppress the vibrations caused by the disturbances beyond the Nyquist frequency.
2. Head-positioning system of hard disk drives A hard disk drive consists of a voice coil motor (VCM), several magnetic heads, several disks, and a spindle motor, as shown in Fig. 1. The head-positioning control system in hard disk drives is illustrated in Fig. 2. In the head-positioning system, the control variable is the head-position signal, which is embedded in disks and is read by magnetic heads. This means the head-position signal is only available as a discrete-time signal. The control input is the voltage supplied to a power amplifier that drives the VCM and magnetic heads, which is calculated by a digital signal processor at certain intervals. Therefore, a head-positioning control system can be thought of as a sampled-data control system that has a sampler and a hold. A block diagram of the sampled-data control system with a multi-rate digital filter is shown in Fig. 3. Here, S is the sampler, C½z is the single-rate digital controller, Ip ½z is the interpolator, F m ½z is the multi-rate digital filter, Hm ðsÞ is the multi-rate hold, Pc ðsÞ is the continuous-time plant, r is the reference signal, ud is the output signal from C½z; uc is the output signal from Hm ðsÞ; dc
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tions and the characteristics of disturbance rejections in a sampled-data system by using frequency responses. The relationship between a continuous-time signal and a discrete-time signal is defined in this paper. For illustration purposes, the sinusoidal wave x0 ðtÞ is defined as
x0 ðtÞ ¼ ejx0 t :
ð1Þ
The discrete-time signal xd ½n, which corresponds to the signal after sampling of x0 ðtÞ with the sampling time T s , can be given as
xd ½n ¼ x0 ðnT s Þ ¼ ejx0 T s n : Fig. 1. Primary components of HDD.
ð2Þ
The continuous-time signal xc ðtÞ, which is created from xd ½n, can be given as
xc ðtÞ ¼
1 X
dðt lT s Þejx0 t ¼
l¼1
1 jðx0 þxs lÞt um1 ; l¼1 e Ts
ð3Þ
where d is a Dirac delta function. Eqs. (2) and (3) indicate that the discrete-time signal caused by a single complex sinusoid can be given by a single frequency, and the continuous-time signal caused by the discrete-time signal consists of multiple frequencies.
3.1. Analysis of sensitivity function
Fig. 2. Head-positioning system.
is the disturbance signal in continuous time, yc0 is the output signal from P c ðsÞ; yc is the head-position signal in continuous time, and yd is the head-position signal in discrete time. In this study, the multirate number is m, the sampling time is T s [s], and the sampling frequency is xs ¼ 2p=T s ½rad=s. The timing used in reading or writing user data in hard disk drives corresponds to intersampling in a sampled-data control system. In other words, the essential purpose of a head-positioning system is to control the maximum displacement of the control variable including intersampling behaviors. Therefore, the control system should be designed as a sampled-data system so that the controller compensates for the intersampling vibrations. The head-positioning system has two control modes; one is ‘‘seek control,” which moves the head onto the target track, and the other is ‘‘following control,” which maintains the head position on the target track. In the seek control, the main target of the control system is the transient characteristics of the head position, but in the following control, the main target is the steady-state characteristics. The purpose of this study is to develop a following control system, and so from here on, the focus of this paper is the steadystate characteristics of the head-position control. 3. Analysis of sampled-data control system using frequency response
In some papers, the gain-frequency response of the sensitivity function in the sampled-data system is defined by using the worst-case H1 norm of systems or the upper/lower bound [24,25]. However, obviously, a head-positioning system has to control the maximum displacement of the control variable, and the maximum displacement has no bound in the steady-state characteristics. Therefore, the infinity norms of signals are used for defining the gain of the sensitivity function for head-positioning control in hard disk drives. In this paper, a gain-frequency response of the sensitivity function in a sampled-data control system jSsd j at x0 is given by the following equation using dc and yc in Fig. 3:
jSsd ðjx0 Þj ¼
Fig. 3. Block diagram of sampled-data control system.
ð4Þ
where dc ðt; x0 Þ is a single complex sinusoid,
dc ðt; x0 Þ ¼ ejx0 t :
ð5Þ
The transfer characteristics from ud to yd at x0 , which is a controlled object in a discrete-time system, is given as Pd ðjx0 Þ,
Pd ðjx0 Þ ¼
1 1 X Wðjx0 þ jxs kÞPc ðjx0 þ jxs kÞ; T s k¼1
ð6Þ
where
WðjxÞ ¼ Ip ½ejxT s =m F m ½ejxT s =m Hm ðjxÞ:
ð7Þ
In a steady-state with dc ðt; x0 Þ ¼ ejx0 t ; ud can be given by the following equation:
ud ½n ¼ To avoid complex calculations involving matrices, the proposed method uses frequency responses to analyze the sampled-data control system [20–23]. The method handles intersampling vibra-
kyc ðtÞk1 ¼ sup jyc ðtÞj; kdc ðt; x0 Þk1 t
C½ejx0 T s ejx0 T s n : 1 þ Pd ðjx0 ÞC½ejx0 T s
ð8Þ
Here, uc ðtÞ and yc0 ðtÞ consist of multiple frequencies and are given by
uc ðtÞ ¼
1 X C½ejx0 T s 1 Wðjx0 þ jxs lÞejðx0 þxs lÞt ; j x T 1 þ Pd ðjx0 ÞC½e 0 s T s l¼1
yc0 ðtÞ ¼
1 X C½ejx0 T s 1 Wðjx0 þ jxs lÞP c ðjx0 þ jxs lÞejðx0 þxs lÞt : 1 þ P d ðjx0 ÞC½ejx0 T s T s l¼1
ð9Þ
ð10Þ
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T. Atsumi / Mechatronics 20 (2010) 67–73
Therefore, yc ðtÞ also consists of multiple frequencies and is given by
yc ðtÞ ¼ yc0 ðtÞ þ dc ðt; x0 Þ 1 X Kðx0 ; lÞejðx0 þxs lÞt ; ¼ Cðx0 Þejx0 t þ
F m ½z ¼ 1 þ F r ½z; ð11Þ
l¼1;–0
CðxÞ ¼ 1
1 C½ejxT s WðjxÞP c ðjxÞ ; T s 1 þ P d ðjxÞC½ejxT s
ð12Þ
1 C½ejxT s Wðjx þ jxs lÞPc ðjx þ jxs lÞ Kðx; lÞ ¼ : Ts 1 þ Pd ðjxÞC½ejxT s
ð13Þ
Finally, jSsd ðjx0 Þj can be given by
t
1 X
jKðx0 ; lÞj:
l¼1;–0
¼
1 X
1þ
!
Kðx0 ; lÞ ejx0 T s n !
Kðx0 ; lÞ ejx0 T s n
1 ejx0 T s n ¼ Sd ðjx0 Þejx0 T s n ; 1 þ P d ðjx0 ÞC½ejx0 T s
5.1. Controlled object
ð15Þ
Pc0 ðjxÞ : jxj 2xs ;
ð23Þ
0 : jxj > 2xs :
Pc0 ðsÞ ¼ Ps ðsÞ Pdl ðsÞ:
ð24Þ
The mechanical model Ps is given by follows [3]:
Ps ðsÞ ¼ K p
w X
ak ðlÞ
l¼1
s2 þ 2fk ðlÞxnk ðlÞs þ xnk ðlÞ2
;
ð25Þ
ð16Þ
Therefore, the control system can suppress the vibration with frequency x0 when
ð17Þ
and 1 X
jWðjx0 þ jxs lÞP c ðjx0 þ jxs lÞj:
20
Gain [dB]
jSd ðjx0 Þj < 1;
jWðjx0 ÞPc ðjx0 Þj
Pc0 is assumed to be given by the product of the ‘‘mechanical model Ps ” and the ‘‘equivalent delay-time model P dl .”
where Sd ðjx0 Þ is the sensitivity function at x0 in a discrete-time system. P1 Eq. (15) indicates that when l¼1;–0 jKðx0 ; lÞj is sufficiently small, Ssd can be given by
Ssd ðjx0 Þ ’ Sd ðjx0 Þ:
The sampling time T s of the hard disk drive used in this study was 76:7 ls (xs ¼ 2p 13000 ½rad=s and the Nyquist frequency was 6500 Hz). The frequency of disturbance xr is set to 7000 Hz. The frequency response of the mechanical system is plotted by the dashed line in Fig. 4.
Pc ðjxÞ ¼
l¼1
¼
ð22Þ
It is assumed that jPc ðjxÞj is vanishingly small when x > 2xs . Thus, to reduce the calculation load, the frequency response of Pc ðjxÞ is defined as
Kðx0 ; lÞejðx0 T s þ2plÞn
l¼1;–0 1 X
Here, the disturbance frequency is defined as xr . To suppress the disturbance, F r should have stable resonant characteristics at xr , and the control system, which includes F r , should satisfy the following equation [26,27]:
5. Control system design
To suppress vibration with frequency x0 , the control system should have a certain level of jWP c j at x0 so that jCðx0 Þj becomes small and a small amount of jWP c j at the aliasing frequencies so P1 that l¼1;–0 jKðx0 ; lÞj becomes small. On the other hand, yd ½n can be given by the following equation:
Cðx0 Þ þ
ð21Þ
ð14Þ
3.2. Disturbance suppression in sampled-data system
1 X
ð20Þ
where
\½Pd ðjxr ÞC½ejxr T s ¼ \½1 þ P d0 ðjxr ÞC½ejxr T s :
l¼1;–0
yd ½n ¼ yc ðnT s Þ ¼ Cðx0 Þejx0 T s n þ
1 1 X W 0 ðjx0 þ jxs kÞP c ðjx0 þ jxs kÞ; T s k¼1
W 0 ðjxÞ ¼ Ip ½ejxT s =m Hm ðjxÞ:
and
jSsd ðjx0 Þj ¼ sup jyc ðtÞj ¼ jCðx0 Þj þ
ð19Þ
where F r is the multi-rate resonant filter. When F r ½z ¼ 0, the transfer characteristics from ud to yd at x0 are given by follows:
Pd0 ðjx0 Þ ¼
where
¼
The multi-rate filter F m is given by the following equation:
0 −20 Measurement data Mathematical model
ð18Þ −40
l¼1;–0
10
3
4
10
Frequency [Hz]
4. Disturbance suppression using multi-rate resonant filter Phase [deg.]
To suppress the disturbance beyond the servo bandwidth of the control system, the control system should have stable resonant characteristics at the disturbance frequency [4,26,27]. However, (18) indicates that the control system should have small gains at aliasing frequencies of the disturbance frequency. Therefore, when the disturbance frequency is higher than the Nyquist frequency, the control system should realize the resonant characteristics by using continuous-time filters or multi-rate filters. In this paper, a design method that uses the multi-rate filter is used.
0 −90 −180 −270
Measurement data Mathematical model
−360 10
3
4
10
Frequency [Hz] Fig. 4. Frequency response of mechanical system.
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T. Atsumi / Mechatronics 20 (2010) 67–73
where w is the number of modes, ak is the residue of each mode, xnk and fk are the natural frequency [rad/s] and the damping ratio of the resonance, respectively, and K p is the plant gain. These parameters were fixed so that the model’s frequency response coincided with the measured result indicated by the dashed line in Fig. 4. To be more precise, w was set to seven (the model order is fourteen), K p to 4:3 107 , and the values of the other parameters are listed in Table 1. The solid line in Fig. 4 represents the frequency response of this mathematical model. The equivalent delay-time model P dl , which includes the characteristics of the delay-time element, is given by follows [20]:
Pdl ðsÞ ¼ ess ;
ð26Þ
where s is the equivalent dead time. Here, used in this study is 10 ls.
s of the hard disk drive
trolled object Pd0 in which F m ½z ¼ 1. The frequency response of Pd0 is indicated by the dashed line shown in Fig. 5. To make the phase margin more than 30°, the gain margin more than 4.5 dB, and the open-loop gain 0-dB crossover frequency 1100 Hz, C is set according to the following equation:
C½z ¼
11:8ðz 0:908Þðz 0:953Þ : ðz 0:161Þðz 1Þ
The frequency response of Pd0 C (the open-loop characteristics without the resonant filter) is indicated by the dashed line in Fig. 6. To suppress the disturbance, the multi-rate filter F m should have stable resonant characteristics at xr . From (19) and (22), the phase of the multi-rate resonant filter F r at the disturbance frequency xr should be
\F r ½ejxr T s =2 ¼ \Pc ðjxr Þ \Ip ½ejxr T s =2 \Hm ðjxr Þ \C½ejxr T s \½1 þ Pd0 ðjxr ÞC½ejxr T s :
5.2. Multi-rate hold and interpolator In this paper, the multi-rate number m is set to two. The multirate hold Hm is a zero-order hold (ZOH) and the transfer characteristic is given by the following equation:
1 eðT s =2Þs : s
ð30Þ
As a result, F m ½z was set as follows:
F m ½z ¼ 1 þ
0:0479ðz 4:37Þðz 1Þ : ðz2 þ 0:233z þ 0:998Þ
ð31Þ
ð27Þ
The interpolator consists of an up-sampler and an interpolation filter. The transfer function of the interpolation filter can be given by
Ip ½z ¼ 1 þ z1 :
ð28Þ
20
Gain [dB]
Hm ðsÞ ¼
ð29Þ
0 −20
Without resonant filter With resonant filter
5.3. Controller design −40
0
1000
2000
First, the single-rate controller C½z, which stabilizes the control system, is given by the ‘‘PI-lead filter” and designed for the con-
3000
4000
5000
6000
5000
6000
Frequency [Hz]
l
xnk ðlÞ
ak ðlÞ
fk ðlÞ
1 2 3 4 5 6 7
0 25,800 35,800 39,000 48,100 55,900 61,600
1.00 1.30 0.03 0.08 0.12 0.13 0.35
0 0.01 0.01 0.01 0.02 0.02 0.03
Phase [deg.]
0 Table 1 Parameters of Ps ðsÞ.
−90 −180 −270 −360
Without resonant filter With resonant filter 0
1000
2000
3000
4000
Frequency [Hz] Fig. 6. Frequency response of open-loop characteristics.
40
Gain [dB]
Gain [dB]
20 0 −20 −40
Without resonant filter With resonant filter 0
1000
2000
20 0
3000
4000
5000
6000
0
Frequency [Hz]
6000
8000
10000
8000
10000
0
−90
Phase [deg.]
Phase [deg.]
4000
Frequency [Hz]
0
−180 −270 −360
2000
Without resonant filter With resonant filter 0
1000
2000
3000
4000
5000
Frequency [Hz] Fig. 5. Frequency response of controlled object.
6000
−90 −180 −270 −360
0
2000
4000
6000
Frequency [Hz] Fig. 7. Frequency response of F m .
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T. Atsumi / Mechatronics 20 (2010) 67–73
jWðjxr ÞPc ðjxr Þj ¼ 3:11;
ð32Þ
15 10 5 0
Gain [dB]
The frequency response of F m is shown in Fig. 7, and the frequency response of P d is indicated by the solid line in Fig. 5. The characteristics of the control system with the resonant filter are explained below. The frequency response of the open-loop characteristics is plotted by the solid line in Fig. 6. The Nyquist diagram and the gain of the sensitivity function in discrete-time system Sd are shown in Figs. 8 and 9, respectively. These figures indicate that (17) is satisfied by using the resonant filter, and (18) is also satisfied because
−5 −10 −15
and 1 X
jWðjxr þ jxs lÞP c ðjxr þ jxs lÞj ¼ 0:11:
−20
ð33Þ
l¼1;–0
−25
By using the design results of the control system and (14), the gainfrequency responses of the sensitivity functions in sampled-data control systems were calculated. The frequency responses of jSsd j
−30 0
2000
4000
6000
8000
10000
12000
Frequency [Hz] Fig. 10. Frequency response of jSsd j.
8
are shown in Fig. 10. As seen from this figure, the control system suppresses the disturbance at 7000 Hz.
6
6. Simulation and experiment
Imaginary axis
4
2
0
−2
−4 −4
−2
0
2
4
6
8
10
Real axis Fig. 8. Nyquist diagram.
The simulation was performed for the control system, in which the frequency of disturbance signal dc is 7000 Hz, and the vibrational amplitude is 0.1 track. In Fig. 11, the dashed line represents the signal of yc without the resonant filter, and the solid line represents the signal of yc with the resonant filter. In Fig. 12, the dashed line represents the signal of yd without the resonant filter, and the solid line represents the signal of yd with the resonant filter. These figures indicate that the control system decreases vibrations in which the frequency is higher than the Nyquist frequency. To validate the effect of the proposed method, we did experiments using the conditions similar to those of the simulation. In these experiments, the sampling time of yc is 38:4 ls (1/2 of T s ) because the continuous-time signal cannot be measured in actual head-positioning systems. The time response of yc is shown in Fig. 13, and the time response of yd is shown in Fig. 14. In both figures, the dashed line represents the results without the resonant filter, and the solid line represents the results with the resonant
15
0.2
10
Without resonant filter With resonant filter
0.15
5 0.1
Position [track]
Gain [dB]
0 −5 −10 −15
0 −0.05 −0.1
−20
−0.15
−25 −30
0.05
0
2000
4000
6000
8000
Frequency [Hz] Fig. 9. Frequency response of jSd j.
10000
12000
−0.2
0
0.5
1
Time [ms] Fig. 11. Simulation results of yc .
1.5
2
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T. Atsumi / Mechatronics 20 (2010) 67–73
filter. These experimental results also indicate that the control system can decrease vibrations whose frequency is higher than the Nyquist frequency. The differences between the simulation result and the experimental result are caused by disturbances other than the disturbance signal dc in the actual head-positioning system.
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05
7. Conclusion
0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
1.5
2
Time [ms]
A control system design method that suppresses disturbances beyond the Nyquist frequency was proposed. This method uses frequency responses of a controlled object and digital controller for calculations of the gain of the sensitivity function in a sampleddata system without complex calculations involving matrices, and realizes a stable resonant filter that decreases the gain of the sensitivity function above the Nyquist frequency. By implementing this method in the head-positioning system of a hard disk drive, it is shown that the control system can significantly suppress the disturbance whose frequency was higher than the Nyquist frequency.
Fig. 12. Simulation results of yd .
References
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
1.5
2
Time [ms] Fig. 13. Experimental results of yc .
0.2 Without resonant filter With resonant filter
0.15
Position [track]
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
0
0.5
1
Time [ms] Fig. 14. Experimental results of yd .
1.5
2
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